Horospheric basis of the SL(2, C) group

Abstract

Representations of theSL(2,C) groups are reduced with respect to the horospheric subgroup$\left( {\begin{array}{*{20}c} 1 \\ 0 \\ \end{array} \begin{array}{*{20}c} \beta \\ 1 \\ \end{array} } \right)$ isomorphic to the translation group in two dimensions. The matrix elements of unitary representations of the principal and supplementary series are derived in a basis defined by a two-dimensional momentum corresponding to the above subgroup. Matrix elements of unitary representations expressed in this basis assume a simpler form than in any other basis considered so far. The two-dimensional momentum introduced proves to be essentially identical with the impact parameter.